The Lorenz curve plots cumulative shares

Greater bowing indicates greater measured inequality.

Order households from lowest to highest income. The horizontal axis records cumulative population share. The vertical axis records cumulative income share. Under perfect equality, the bottom 40 percent receives 40 percent of income, producing the 45-degree line. An actual Lorenz curve lies below it. Farther distance indicates greater inequality when curves do not cross.

If two Lorenz curves cross, neither distribution is unambiguously more equal by the Lorenz criterion. One may give a larger share to the bottom group while the other gives a larger share to the middle. A single point cannot rank entire curves unless the rest of the relationship is known.

A Lorenz curve converts an entire income distribution into cumulative shares. First rank households from lowest to highest income. The horizontal axis shows the cumulative percentage of households. The vertical axis shows the cumulative percentage of income they receive. Every valid curve begins at (0,0) and ends at (100,100) because zero households receive zero income and all households receive all income.

The 45-degree line represents perfect equality: the bottom 20 percent receives 20 percent of income, the bottom 60 percent receives 60 percent, and so on. An actual Lorenz curve normally bows below that line. If the bottom 40 percent receives only 15 percent of income, plot the point (40,15). Greater bowing means a larger cumulative gap from equality.

In this table, A gives larger cumulative shares through 60 percent, but B gives a larger share through 80 percent. The curves cross, so Lorenz dominance cannot rank one as more equal at every cumulative population share. A single point comparison is valid only at that point.

Do not confuse cumulative and group shares. If the bottom 20 percent receives 5 percent and the bottom 40 percent receives 16 percent, the second quintile alone receives 16-5=11 percent. The top 20 percent’s share equals 100 minus the cumulative share received by the bottom 80 percent. These subtractions are frequent numerical traps.

The curve describes relative distribution, not living standards. If every household’s income doubles, the cumulative shares and Lorenz curve remain unchanged even though material resources rise. Two countries can share the same curve while one has much higher income. Wealth, annual earnings, pretax household income, and after-tax income can also yield different curves, so comparisons require consistent definitions.

Taxes and transfers can move the after-tax Lorenz curve toward equality, while demographic change or recession can alter the measured distribution in complicated ways. The curve does not reveal why inequality exists or whether it is fair. It is a positive descriptive tool. Normative evaluation requires additional values and attention to incentives, mobility, and absolute well-being.

To construct a curve from quintile shares, cumulate rather than plot each group’s stand-alone percentage. If successive quintiles receive 5, 10, 15, 25, and 45 percent, the plotted income coordinates are 5, 15, 30, 55, and 100. Those values must never decline because adding another household group cannot reduce cumulative income.